1720, "woman accompanying a younger, unmarried lady in public," from French chaperon "protector," especially "female companion to a young woman," earlier "head covering, hood" (c.1400), from Old French chaperon "hood, cowl" (12c.), diminutive of chape "cape" (see cap (n.)). "... English writers often erroneously spell it chaperone, app. under the supposition that it requires a fem. termination" [OED]. The notion is of "covering" the socially vulnerable one.

"May I ask what is a chaperon?"
"A married lady; without whom no unmarried one can be seen in public. If the damsel be five and forty, she cannot appear without the matron; and if the matron be fifteen, it will do."
[Catharine Hutton, "The Welsh Mountaineer," London, 1817]

The word had been used in Middle English in the literal sense "hooded cloak."